The Failure of Rank-One Connections

Iwaniec, Tadeusz; Verchota, Gregory C.; Vogel, Andrew

doi:10.1007/s002050200197

Cited by 35 publications

(32 citation statements)

References 15 publications

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“…Nevertheless we are very far from understanding how to solve it in any generality, one obstacle being the well-known lack of a useful characterization of quasiconvexity (see, for example, [22]), which is known to be a key to understanding compatibility. The generalizations of the Hadamard jump conditions considered in [4] (see also [23]) are also insufficiently general and tractable. As well as for polycrystals, such generalized jump conditions are potentially relevant for the analysis of nonclassical austenite-martensite interfaces as proposed in [24,25], which have been observed in CuAlNi [26,27], ultra-low hysteresis alloys [28], and which have been suggested to be involved in steel [29].…”

Section: Theorem 5 ([4])mentioning

confidence: 99%

Geometry of polycrystals and microstructure

Ball

Carstensen

2015

MATEC Web of Conferences

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Abstract. We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-totetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.

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Section: Theorem 5 ([4])mentioning

confidence: 99%

Geometry of polycrystals and microstructure

Ball

Carstensen

2015

MATEC Web of Conferences

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“…(ii) For general results concerning the relationship between the gradients of a Lipschitz mapping on either side of an interface see Ball & Carstensen [3], Iwaniec et al [12].…”

Section: Notationmentioning

confidence: 99%

An investigation of non-planar austenite–martensite interfaces

Ball¹,

Koumatos²

2014

Math. Models Methods Appl. Sci.

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Abstract. Motivated by experimental observations on CuAlNi single crystals, we present a theoretical investigation of non-planar austenite-martensite interfaces. Our analysis is based on the nonlinear elasticity model for martensitic transformations and we show that, under suitable assumptions on the lattice parameters, non-planar interfaces are possible, in particular for transitions with cubic austenite.

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“…Therefore, at almost every point z ∈ X we either have h z = 0 or hz = 0 . The interested reader is referred to [30,34,35] for more details. Thus we have a nonzero solution of the homogeneous Dirichlet problem…”

Section: To This Effect Note That Rementioning

confidence: 99%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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Abstract. The paper is concerned with mappings h : X onto − − → Y between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X ) of the energyminimal mappings is established within the class H 2(X, Y) of strong limits of homeomorphisms in the Sobolev space W 1,2 (X, Y) , a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation (of the independent variable in X) leads to the Hopf differential hzhz dz ⊗ dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle:A mapping h ∈ H 2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in X . Nevertheless, cracks are triggered only by the points in ∂Y where Y fails to be convex. The general law of formation of cracks reads as follows:Cracks propagate along vertical trajectories of the Hopf differential from ∂X toward the interior of X where they eventually terminate before making a crosscut.

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The Failure of Rank-One Connections

Cited by 35 publications

References 15 publications

Geometry of polycrystals and microstructure

Geometry of polycrystals and microstructure

An investigation of non-planar austenite–martensite interfaces

Mappings of Least Dirichlet Energy and their Hopf Differentials

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